No splitter widget in new separate window
The reason you cannot split the windows is because you have one of your part of the User Interface maximized. 



Once you go out of this maximization via pressing the "Back to previous" in the info header or by pressing Shift Space you can change your UI and create new windows.

Some other shortcuts to manipulate the User Interface.


Shift+Space, Ctrl+Up,Ctrl Down: Maximize Window under mouse
Ctrl+Left, Ctrl+Right: go to previous/next Layout